Let I = [0, 1] be the compact topological semigroup with max multiplication and usual topology. C(I), $L^p\left(I\right)$, 1 ≤ p ≤ ∞, are the associated Banach algebras. The aim of the paper is to characterise $Ho{m}_{C\left(I\right)}(L^r\left(I\right),L^p\left(I\right))$ and their preduals.

We consider a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions. We show that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurcate from the branch of trivial solutions. This bifurcation point lies in the domain of stability of the trivial solution to the same system with Dirichlet and Neumann boundary conditions, where a bifurcation of this classical problem is excluded.

This paper uses minimization methods and renormalized functionals to find spatially heteroclinic solutions for some classes of semilinear elliptic partial differential equations

This paper uses minimization methods and renormalized functionals to find spatially heteroclinic solutions for some classes of semilinear elliptic partial differential equations

For the difference equation $\left(ϵ\right)x_{n+1}=A{x}_n+ϵ{\sum }_{k=-\infty }^n{R}_{n-k}{x}_k$,where $x_n\in Y,Y$  is a Banach space, $ϵ$ is a parameter and  $A$  is a linear, bounded operator. A sufficient condition for the existence of a unique special solution  $y={\left\{y_n\right\}}_{n=-\infty }^{\infty }$  passing through the point  $x_0\in Y$  is proved. This special solution converges to the solution of the equation (0) as  $ϵ\to 0$.

Toeplitz operators on strongly pseudoconvex domains in Cn, constructed from the Bergman projection and with symbol equal to a positive power of the distance to the boundary, are considered. The mapping properties of these operators on Lp, as the power of the distance varies, are established.

Let Ω= [a, b] × [c, d] and T 1, T 2 be partial integral operators in $$C$$ (Ω): (T 1 f)(x, y) = $$\underset{a}{\overset{b}{\int }}$$ k 1(x, s, y)f(s, y)ds, (T 2 f)(x, y) = $$\underset{c}{\overset{d}{\int }}$$ k 2(x, ts, y)f(t, y)dt where k 1 and k 2 are continuous functions on [a, b] × Ω and Ω × [c, d], respectively. In this paper, concepts of determinants and minors of operators E−τT 1, τ ∈ ℂ and E−τT 2, τ ∈ ℂ are introduced as continuous functions on [a, b] and [c, d], respectively. Here E is the identical operator in C(Ω). In addition, Theorems on the spectra of bounded...

The essential spectrum of bundle shifts over Parreau-Widom domains is studied. Such shifts are models for subnormal operators of special (Hardy) type considered earlier in [AD], [R1] and [R2]. By relating a subnormal operator to the fiber of the maximal ideal space, an application to cluster values of bounded analytic functions is obtained.

We give a simple proof of the relation between the spectra of the difference and product of any two idempotents in a Banach algebra. We also give the relation between the spectra of their sum and product.

Let $E$ be a complex Banach space, with the unit ball $B_E$. We study the spectrum of a bounded weighted composition operator $u{C}_{\varphi }$ on $H^{\infty }\left(B_E\right)$ determined by an analytic symbol $\varphi$ with a fixed point in $B_E$ such that $\varphi \left(B_E\right)$ is a relatively compact subset of $E$, where $u$ is an analytic function on $B_E$.

Burgos, Kaidi, Mbekhta and Oudghiri [J. Operator Theory 56 (2006)] provided an affirmative answer to a question of Kaashoek and Lay and proved that an operator F is of power finite rank if and only if ${\sigma }_{dsc}(T+F)={\sigma }_{dsc}\left(T\right)$ for every operator T commuting with F. Later, several authors extended this result to the essential descent spectrum, left Drazin spectrum and left essential Drazin spectrum. In this paper, using the theory of operators with eventual topological uniform descent and the technique used by Burgos et...